With the enlightment of a reply a few edits and additions.

This is to just remove the cowsay ascii characters but still keeping the random fortunes.

To remove the cow ascii but to preserve the file just in case you have a change of heart:

Open a terminal (CTRL+ALT+T) if you haven't already.
Type in: sudo gedit /usr/bin/mint-fortune
Find the line at the bottom that looks like:
/usr/games/fortune | $command -f $cow
Make it look like:
#/usr/games/fortune | $command -f $cow
then put this at the bottom:
/usr/games/fortune

SAVE

Thats it no more cow ASCII, and if you ever want it back to the way it was just undo what you did.

To maybe have a more permanent change but not a guarentee you could instead do the following:

Open a terminal (CTRL+ALT+T) if you haven't already.
Type in: sudo gedit /etc/bash.bashrc
Find the line at the bottom that looks like:
/usr/bin/mint-fortune
Make it look like:
#/usr/bin/mint-fortune
then put this at the bottom:
fortune

SAVE

Again, no more cow ASCII, and if you ever want it back to the way it was just undo what you did.

Be aware however an update may also overwrite that file as well giving you back your cow ascii.

Some of you might think why not just remove cowsay, I am one of those people. However if you remove cowsay you also remove three other packages (mint-artwork-common, mint-artwork-gnome, and mintsystem) and as of right now I am not 100% sure you want to do that as it might mess up your panels among a few other things.